MIT15_053S13_iprefguide

Suppose that we want to select a minimum cardinality

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Unformatted text preview: he counties of Colorado. Figure 1. The counties of Colorado. Suppose that we want to select a minimum cardinality subset C of counties such that each county either was in C or shared a common boundary with a county in C. In this case, we would let 5j = {j} U {i: county i shares a boundary with county j}. We let Cj = 1 for each j. We would then solve the min cost covering problem. Here is a related problem. Select the largest subset CC of counties such that no two counties of CC share a common border. We refer to this problem as P1. Although P1 is a set packing problem (as you shall soon see) it is not the set packing problem defined on the sets S1, S2, ", Sn . To see why not, consider the following example. Suppose that county 1 bordered on county 2, which bordered on county 3, but that counties 1 and 3 had no common border. Then one can include both county 1 and county 3 in CC. But 2 E S1 n S3, and so it would not be possible to select S1 and S3 for the set packing problem. Problem P1 can be formulated as follows: Maximize ∑ subjec...
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